Ratio & Proportion

Ratio and Proportion

Ratio is a means to compare like quantities in terms of how many.

Comparing like Quantities

Comparing quantities means finding if one quantity is the same as, greater than or smaller than the other quantity.

Class 6 Maths :

Comparison by Difference – by how much?

Difference of two like quantities tells us that one quantity is how much bigger or smaller than the other quantity.
e.g., Cost of a car is Rs 250000 and that of a motorbike is Rs 50000,

Cost of car − Cost of bike = Rs (250000 − 50000) = Rs 200000

We say that car is costlier than bike by = Rs 200000.

Comparison by Division – by how many times ?

Division of two like quantities tells us that one quantity is how many times of the other.

If we compare the above example by division,

Cost of car / Cost of bike = \frac{250000}{50000} = \frac{5}{1}

We say that car is 5 times costlier than bike.

Same unit cancel each other out during division and the result we get is number.

❕ Measurement of a physical quantity involves comparison with a reference standard called unit.

Measure of a quantity = n × unit of the quantity

Where, n = number of times a measured quantity is greater or smaller than the unit quantity.

e.g., If a subtance weighs 5 Kg then this means that it is 5 times heavier than the reference standard of 1Kg.
If a person is 155 cm tall then this means that he is 155 times taller than the reference standard of 1cm.

Ratio

“Comparison of quantities in terms of ‘how many times’, is known as Ratio.”

● Ratio is the comparison by division.

● We denote ratio by using symbol : .

Ratio of x to y,
\boxed{x\colon y \implies \frac{x}{y}}

In earlier example, the ratio of cost of car to the cost of car = \frac{250000}{50000} = \frac{5}{1} = 5 : 1

● Two quantities can be compared by ratio only if they are of same type and in the same unit.

i.e., two lengths can be compared only when they are in same unit e.g., cm to cm, m to m etc.

ℹ [Rate: It is a special ratio which compare different kinds of quantities or units. The word “per” is used instead of “to”. e.g., miles per hour (speed), miles per litre (mileage), price per kg (cost) etc.]

● The order in which quantities are taken to express their ratio is important, the ratio 3 : 2 is different from 2 : 3.

Equivalent Ratios

Ratios which are same in their lowest form are called equivalent ratios.

e.g., \frac{30}{20} = 3 : 2 , \frac{24}{16} = 3 : 2 .

∴ 30 : 20 and 24 : 16 are equivalent ratios.

● We can get equivalent ratios by multiplying or dividing the numerator and the denominator of a ratio by same number.

● It is same like equivalent fractions.

e.g., 9 : 6 = \frac{9\times 2}{6\times 2} = \frac{18}{12} = 18 : 12

∴ 18 : 12 is an equivalent ratio of 9 : 6

also, 9 : 6 = \frac{9\div 3}{6\div 3} = \frac{3}{2} = 3 : 2

So, 3 : 2 is another equivalent ratio of 9 : 6.

Try These

1. In a class, there are 20 boys and 40 girls.

(a) What is the ratio of the number of boys to the number of girls?

(b) What is the ratio of the number of girls to the number of boys?

(c) What is the ratio of the number of boys to the total number of students?

2. Saurabh takes 15 minutes to reach school from his house and Sachin takes one hour to reach school

from his house. Find the ratio of the time taken by Saurabh to the time taken by Sachin.

3. Divide Rs 60 in the ratio 1 : 2 between Kriti and Kiran.

Answer

1. Number of boys = 20

Number of girls = 40

Total number of students = 20 + 40 = 60

(a) Ratio of the number of boys to the number of girls = 20/40 = 1 : 2

(b) Ratio of the number of girls to the number of boys = 40/20 = 2 : 1

(c) Ratio of the number of boys to the total number of students = 20/60 = 1 : 3

2. Here, times are in different units, to compare them we must convert them to same unit.

Time taken by Saurabh to reach school = 15 minutes

Time taken by Sachin to reach school = 1 hour = 60 minutes

∴ Ratio of the time taken by Saurabh to the time taken by Sachin = 15/60 = 1 : 4

3. Total part in which Rs 60 is divided = 1+2 = 3

Kriti gets 1 part of it i.e., = 1/3 of Rs 60 = Rs \frac{1}{3}\times 60 = Rs 20

Kiran gets 2 part of it i.e., = 2/3 of Rs 60 = Rs \frac{2}{3}\times 60 = Rs 40

We can check our result,

Ratio of the amount Kriti gets to the amount Kiran gets = \frac{20}{40} = \frac{1}{2}

Hence, our answer is correct.

Proportion

If two ratios are equal, we say that they are in proportion and use the symbol ‘ :: ’ or ‘ =to equate the two ratios.

● 3, 10, 15 and 50 are in proportion which is written as 3 : 10 :: 15 : 50 or 3 : 10 = 15 : 50, and is read as 3 is to 10 as 15 is to 50.

ℹ Proportion has many applications, some of which are scale drawings, maps, constructions etc.

To make a drawing accurately, the dimensions in drawing must be in proportion with the dimensions in original i.e., ratio of different lengths/curves in drawing must be equal to the ratio of respective lengths/curves in original.

● National flags are always made in a fixed ratio of length to its breadth. These may be different for different countries but are mostly around 1.5 : 1 or 1.7 : 1.

Respective Terms

In a statement of proportion, the four quantities involved when taken in order are known as respective terms.

First and fourth terms are known as extreme terms.

Second and third terms are known as middle terms.
Product of extremes = Product of means

If a, b, c and d are in proportion, then

\boxed{a\, \colon b\, \colon\colon c\, \colon d}

\boxed{a\times d = b\times c}

● If two ratios are not equal, then they are not in proportion.

● The order of terms in the proportion is important e.g., 2, 5, 6 and 15 are in proportion, but 2, 5, 15 and 6 are not, since 2:5 is not equal to 15:6.

Try These

Check whether the given ratios are equal, i.e. they are in proportion. If yes, then write them in the proper form.

1). 1 : 5 and 3 : 15

2). 2 : 9 and 18 : 81

3). 15 : 45 and 5 : 25

4). 4 : 12 and 9 : 27

5). Rs 10 to Rs 15 and 4 to 6.

Answer

1) 1 : 5 = 1/5 and 3 : 15 = 3/15 = 1/5

∴ Ratios are equal, i.e. they are in proportion, 1 : 5 :: 3 : 15

2) 2 : 9 = 2/9 and 18 : 81 = 18/81 = 2/9

∴ Ratios are equal, i.e. they are in proportion, 2 : 9 :: 18 : 81

3) 15 : 45 = 15/45 = 1/3 and 5 : 25 = 5/25 = 1/5

∴ Ratios are not equal, i.e. they are not in proportion.

4) 4 : 12 = 4/12 = 1/3 and 9 : 27 = 9/27 = 1/3

∴ Ratios are equal, i.e. they are in proportion, 4 : 12 :: 9 : 27

5) Rs 10 : Rs 15 = 10/15 = 2/3 and 4 : 6 = 4/6 = 2/3

∴ Ratios are equal, i.e. they are in proportion, Rs 10 : Rs 15 :: 4 : 6

Problem Solving

Unitary Method

“The method in which first we find the value of one unit and then the value of required number of units, is known as Unitary Method.”

ℹ In this method, we first found the value for one unit or the unit rate . This is done by comparing the properties (or quantities) by divison.

● We often use per to mean for each. e.g., km per hour, rupees per kg etc., denote unit rates.

● Then we multiply the unit rate with required number of units whose value is to be found.

Ex- If the cost of 6 cans of juice is Rs 210, then what will be the cost of 4 cans of juice?

A) Given: The cost of 6 cans of juice = Rs 210

The cost of 1 cans of juice (unit rate) = Rs \frac{210}{6} = Rs 35

Then, The cost of 4 cans of juice = Rs 35 × 4 = Rs 140

∴ The cost of 4 cans of juice is Rs 140.

Ex- Cost of 4 dozens bananas is Rs 60. How many bananas can be purchased for Rs 12.50?

A) Given: In Rs 60, no of bananas that can be purchased = 4 dozens = 4×12 = 48

In Re 1, no of bananas that can be purchased =\frac{48}{60} =\frac{4}{5}

In Rs 12.50, no of bananas that can be purchased = \frac{4}{5}\times 12.50

= 4 × 2.5

= 10

∴ 10 bananas can be purchased for Rs 12.50.

Algebraic Method

➣ First we assign a variable x to the quantity to be found .

➣ Then, put it in an equation according to the condition.

➣ Solve the equation for x, which is the required answer

Ex- In a computer lab, there are 3 computers for every 6 students. How many computers will be needed for 24 students?

A) Let, the computers needed for 24 students be x.

A/Q (According to the question)

\frac{3}{x} = \frac{6}{24}

⇒ 3×24 = x×6 (cross-multiplication)

⇒ 72 = 6x

⇒ 6x = 72 (re-arranging)

⇒ x = 72/6

⇒ x = 12

∴ The computers needed for 24 students = x = 12.

Problems On Ratio And Proportion

You can check problems on topics of Ratio And Proportion by clicking the links given below.

Problems on Ratio
Problems on Proportion
Problems on Unitary Method

We hope this article has helped you understand Ratio, Proportions and related concepts.

Write in the comment section for any error or any related queries from the exercise.

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